The Erd\H{o}s-Ko-Rado basis for a Leonard system

TL;DR

This paper introduces an Erd ext{"o}s-Ko-Rado basis for Leonard systems, generalizing linear programming methods used in classical Erd ext{"o}s-Ko-Rado theorems to a broader algebraic framework.

Contribution

It defines a new basis for Leonard systems satisfying certain eigenvalue conditions, expanding the algebraic tools for combinatorial and graph-theoretic applications.

Findings

Describes transition matrices between bases

Provides matrix representations of A and A* in the new basis

Generalizes linear programming approach for Erd ext{"o}s-Ko-Rado theorems

Abstract

We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi = (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^* \}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^*$. We describe the transition matrices to/from other known bases, as well as the matrices representing $A$ and $A^*$ with respect to the new basis. We also discuss how these results can be viewed as a generalization of the linear programming method used previously in the proofs of the "Erd\H{o}s-Ko-Rado theorems" for several classical families of $Q$-polynomial distance-regular graphs, including the original 1961 theorem of Erd\H{o}s, Ko, and Rado.